Well, this is an awkward question and I don't know if it is mathematical enough for MO (I'm sorry if not) but I'll try it: What observations in the coordinate system centered in my fixed position on earth are necessary to conclude that the earth (and the planets) move (approximately) in ellipses around the sun and that earth is rotating around itself?

Thumbs down to the votes to close! Boooo.
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Kevin H. LinJul 10 '10 at 21:26

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Uh,looking up in the daytime is a rather unambigous clue......LOL Just kidding. It's actually an excellent question since all of us grew up at a time in history when it's more or less taken as obvious.Considering it from first principles,it really isn't. Also,there's so much mythology and apocrophya surrounding the gradual development of the heliocentric model that culminated in classical mechanics,that most of us nonexperts simply have erroneous ideas of how it was actually proven by the Ancient Greeks and thier sucessors.
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Andrew LJul 11 '10 at 1:01

Ernst Mach would have claimed that you get the same result if the earth stood still and the stars were rotating. Without assuming certain basic laws of physics (Newton, Einstein), it seems that you cannot deduce anything at all.
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Franz LemmermeyerFeb 7 '10 at 7:10

I think that you can write down the motion of planets in any coordinate system you like. Since prehistoric times, astronomers have recorded the motion of planets, and before Kepler everybody used the coordinate system where the earth is fixed. I don't know if they ever wrote down equations of the planets' motion, but they could certainly predict it.

Of course, the equations of motion in this coordinate system would be very messy and inconvenient. The reason that we say that "planets move around the sun" is, IMHO, the fact that in the coordinate system centered at the sun, the equations become so simple and easy to understand.

The one "specific observation" that, to my mind, shows how confusing and inconvenient to calculate the motion of planets is from an Earth-centered system, is the retrograde motion of planets. When viewed from the earth, planets usually move in one direction in the sky (when viewed at the same time but on different days). Sometimes, however, planets like Mars move backwards in the sky. I think this has to do with the angular velocity of the Earth moving around the sun being greater than that of Mars (anybody has a better explanation?).

I believe that is the correct reason, Ilya.
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Harry GindiFeb 6 '10 at 22:56

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@Harry: Here's what confuses me: why doesn't Mars then move backwards always? @Arminius: Next, you try to write down equations to predict the exact motion, and try to find the coordinate system in which it's easiest. I'm not sure if there are any qualitative phenomena that will guide you, although if you invent some other "wrong" coordinate system, there will probably be something qualitatively weird with it. To get it right, you might need Kepler-level intelligence.
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Ilya GrigorievFeb 6 '10 at 23:06

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@Arminius: Of course, there might be a better answer to your question, especially if we instead ask: how did Kepler justify his theory? What exact observations did he use? But I don't know, and I suspect that even if you study this in detail, you might not find an answer that will satisfy you exactly. It's easy to explain why a theory is good, but very hard to explain how to make up a good theory.
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Ilya GrigorievFeb 6 '10 at 23:09

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@Ilya " What exact observations did he use?" --- he used observations of Mars orbit by Tycho Brage. @Arminius I suggest you to read the Keplers book "Astronomia nova". Answer should be there.
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PetyaFeb 6 '10 at 23:36

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@Ilya: Sorry, I did not see your comment until after I had posted my comment. To answer your question: Yes, it is certain that Copernicus was aware of Aristarchus's priority because the original draft of his 1543 book has survived and it included a passage that refers to Aristarchus, which Copernicus later crossed out so as not to compromise the originality of his theory.
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Marko AmnellMar 17 '10 at 18:02

Evidence that the Earth is spinning about its axis: the Coriolis effect. The coordinate system fixed to a specific spot on the Earth's surface is not inertial, that is, Newton's first law does not hold.

Evidence that the Earth and other planets are moving about the Sun in elliptical orbits: agreement of the projected elliptical motions with observations in the night sky.

There is no fundamental reason to choose the Sun as the center of the solar system coordinates. Any point will do, including the Earth or any other planet. However, as noted by such luminaries as Copernicus, Galileo, Kepler and other MO respondents, choosing the Sun as the center simplifies things quite a bit.

I think Copernicus' role in all this is a bit more complicated than is often assumed -- I attended several history of mathematics lectures some years ago, which argued in semi-jocular but also semi-serious fashion that his heliocentrism was mystical and not scientific. But good answer, nonetheless.
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Yemon ChoiFeb 7 '10 at 0:02

I find the parallax effect
parallax
effect especially convincing evidence. Parallax is the shifting of lines of sight
due to translation, eg by waiting half an earth year at which point
theory tells we have moved about 16 light minutes around the sun from where we were.

Regarding retrograde motions: as Ilya said,
Kepler's 2nd law closer
planets move faster. Now draw two circles
centered at 'Sun' with a point 'Earth' moving on the inner circle
and another point 'Mars' moving more slowly but in the same sense, say counterclockwise
on the outer circle. Drow a line between the two moving points. That line
indicates how Mars looks, viewed from earth, relative to the distant stars.
How does the line move? Put the Sun at the origin. If the order is Sun-Earth-Mars,
with Earth and Mars on the positive
x axis, then the slope of the line is decreasing. But put the order Earth-Sun-Mars
with Earth on the negative x axis, Mars on the positive x -axis. The slope of said line is now increasing. One is `prograde' the other 'retrograde'.

Finally, the explanation of elliptical versus circular motion
is more of an Occam's razor business. Originally we had
Ptolemy's ''epicycles''-- in essence Fourier series. Ptolemy had earth at the
solar system center and each planet moving on a system of nested circles,
as in $z(t) = r_1 e ^{i \omega_1 t} + r_2 e^{i \omega_2 t} + r_3 e^{i \omega_3 t} + ... $, $r_1 > r_2 > \ldots $. Ptolemy needed 20 to 30 circles to account for observations.
Kepler realized that but putting the sun at the 'center' and having the planets move in slightly eccentric ellipses with focus,
a bit off from the sun, sweeping out ''equal areas in equal times'', he could account for all of Ptolemy's data plus Brahe's much more detailed data.

An interesting (and very old) argument in favour of heliocentrism is based on estimates of the relative sizes of the Earth and Sun.

Actually, Aristarchus of Samos estimated that the Sun is six to seven times wider than the Earth (and therefore over 200 times more voluminous). These calculations arguably led him to conclude that it made more sense for the Earth to be moving than for the huge Sun to be moving around it.

That's how I learned it was suggested-although it took nearly another 2000 years for it to become the dominant worldview once Aristotle's ideas began to take hold in the Western World,which favored geocentrism. The influence of the Christian Church further reinforced this. Something else we can thank the Church for........
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Andrew LJul 11 '10 at 1:06

@ Andrew: there are plenty of other "churches", including the lack of any, to be thanked for many uglier things. Comments on this subject are not math related and I think should not belong here.
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Yaakov BaruchJul 11 '10 at 11:08

All the answers given are essentially correct - motion in inertial and non-inertial coordinate systems is governed by quite different laws, which are pretty easy to observe. Recall, that Newton's laws (at least the first two) hold only in inertial coordinate systems.

I just wanted to point out that this is a fundamental reason why we choose the Sun as a center - this coordinate system is more inertial. Not just because "equations look simpler". "Occam's razor business" also has nothing to do with this.

Question itself is a bit anachronic - people knew this many centures ago.

Stellar aberration, the change in the apparent positions of more or less most of the night sky with the seasons, directly due to the velocity of the earth in its orbit. Note - aberration, which is related to the idea that raindrops appear to fall slanted in a moving vehicle, is not the same as parallax, e.g. has no relation to the distance of the stellar source, etc. :-)
See http://en.wikipedia.org/wiki/Aberration_of_light .
Predicted by Bradley in 1725. First measured > 0 by Bessel in 1838, with unpublished successful observations by Henderson 5 years earlier.
See http://thonyc.wordpress.com/2010/01/09/we-know-it-moves-but-can-you-prove-it/ .

Isn't the whole idea why we accept the 'fact' that the Earth revolves around the sum based on how science works?

You have a theory about how something works, do some experiments, and if you don't get a contradiction, you do not reject the theory. I think that if you investigated any of the theories that placed the Earth at the center of the Universe, you would eventually find a contradiction. When you place the Sun at the center of the Solar System, no contradictions develop ( except for the thing about the behavior of Mercury - ask Eddington and Einstein).

Actually, when you listen to phyicists about why they accept quantum mechanics, they will say that it is a theory that has never encountered a contradiction over the past eighty years. However, they are still looking to a theory to unit it with general relativity, but that is another story.